When attempting to reach the exact ground states of interacting quantum systems, the computational cost increases exponentially with the system size in most cases and is thus intractable, which is known as an NP-hard problem. Unless accurate quantum computers become available, approximate but sufficiently accurate algorithms that are applicable to conventional computers need to be developed. This demand is particularly strong in research fields, such as condensed matter, high-energy physics, and chemistry.

Various methods have been developed to achieve these objectives. Recently, artificial neural networks based on machine learning have demonstrated promising performances. For instance, they contribute to establish the existence of a quantum spin liquid that embodies strong long-range quantum entanglement, which is expected to be useful in quantum computers and information transmission.

However, the existing neural networks employ classical variables, such as Ising spins for hidden variables in Boltzmann machines, to represent the interaction and entanglement effects. As quantum entanglement is not efficiently handled by classical operations, hidden quantum variables that efficiently handle long-range quantum entanglement are desired to be incorporated.

The question then is how to build an architecture that incorporates hidden quantum variables. A hint was provided by a recent understanding of strongly correlated electrons, particularly achieved by machine-learning analyses of spectroscopic experimental data and ab initio calculations of cuprate high-T_c superconductors. The key understanding is that electrons in the cuprates behave as if the original single electron is splintered into two noninteracting fermions, dubbed “electron fractionalization”, a remarkable but ubiquitous feature if bistability exists, as in examples near the first-order transitions.

A striking idea from this fractionalization is that strongly correlated fermions may be mapped to noninteracting systems, where the fractionalization is embodied by mutually hybridizing multiple components of fermions. As treating strong interaction and many-body entanglement is a widely recognized grand challenge, it would be astonishing if it is mapped to tractable noninteracting systems. One might be suspicious about this “free lunch”.

In this study, by taking the Hubbard model, known as a simple but challenging model of strongly correlated electrons, the exact mapping of the full Hilbert space of one- and two-site Hubbard models, to a two-component noninteracting fermion model (TCFM) is established. An extension of the mapping to larger systems and the architecture of the noninteracting multicomponent fermion model (MCFM) are proposed to simulate an any-sized Hubbard model, which enables a new quantum many-body solver called the Fermi machine. The Fermi machine easily treats quantum entanglement through the hybridization of hidden fermions and physical electrons, and easily reproduces the Mott insulator as well as the pseudogap states in cuprate superconductors by hybridization gaps.

The algorithm was tested for the four-site Hubbard model, and it easily reproduced the exact ground state by the MCFM.  The Fermi machine has significant potential for accurate and functional quantum many-body solvers by utilizing the nature and structure of strongly correlated real electrons. The concept of mapping between strongly interacting quantum particles and tractable noninteracting quantum systems will provide the basis for a deeper understanding of the interacting quantum world.

(Written by Masatoshi Imada)

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